taeto

Question: is the problem to find the set \(\{a,b,c,d\}\) of digits, or the actual 4-tuple \((a,b,c,d)\) (i.e. the 4-digit number "abcd")?

It is impossible to deny the 4th law of motion, because its authoritative webpage also has a depiction of the flying spaghetti monster: www.quora.com/What-is-Newton’s-fourth-law-of-motion-which-was-a-hidden-mystery

However interesting, this isn't about a theorem or anything. Maybe it will get more attention if moved to Brain Teasers and Puzzles.

Epsilon is a symbol, a Greek letter. It only becomes a name if it names something. In a text on calculus or analysis the author will explain that you will have to think of a fixed arbitrary positive real number and give it the name epsilon. That makes it possible for the reader to figure out what kind of thing the epsilon is the name of. You have not done anything like that. Usually this means by default that in your presentation epsilon also has to mean a positive real number.

There are many mistakes in the paper. The answer to the question whether the proof is correct is therefore negative; it is not correct. A correct proof does not contain several mistakes. But still, assuming that it is possible to get a rough understanding of the idea, we can consider how much sense it makes. The conjecture, as it is generally understood, suggests that every even number greater than 2 is a sum of two primes. The paper instead considers the version in which every even number greater than 4 is required to be expressed as a sum of two odd primes. The conjecture applies to all even numbers from 6 and higher. If the idea of the proof is correct, it should show for every even number \(N\) greater than or equal to 6 that there are two odd primes \(p,q\) with \(p+q=N.\) If it does not do this, then the idea is not correct. Do you see how the proof shows for \(N=6\) that the primes \(p\) and \(q\) exist? More precisely, in expression [9], what are the values of \(p_1,p_2,p_3,p_4,p_5,p_6\) that make the proof work in this case?

Not quite. In [9], [10] and [11] we have inequalities \(\neq.\)

When we compare [10] and [11] it only makes sense if we assume \(p_7+p_8=2.\)

My first reaction, and I have not been in a chem lab for ages, is that the shape of the liquid is generally going to be as in figure A. It certainly is for water. The lower layer provides the most accurate measurement, since it is usually much wider than the upper narrow portion. Since water is transparent, you can observe the height of the lower layer from the side, which gives the optimal value. If a liquid is not transparent, you cannot see the lower layer, and you have to measure the height of the upper layer. I ready myself for being made fun of for a simpleton explanation of something that is probably infinitely more subtle.

But the author of http://new-idea.kulichki.net/pubfiles/190210115350.pdf chose the equivalent wording: Each odd number greater than 5 can be represented as a sum of three primes. Are you discussing this paper, or a different paper?

Why would you say that? The equation [1] clearly talks about the weaker conjecture, which does not require odd primes. Or am I looking at the wrong version of the paper now? Please replace it by the correct version.

I hear you loud and clear. My remark was only circumferential. Diagrams like yours might provide OP with a good aid to understanding more about presentation, so kudos for that.

Nice. But someone here will argue that your intuition about the (basic?) concept of "circumference" of a circle is off by a tiny amount.

Personally I think that the use of the same words which are already used in classical calculus, but now with entirely different meanings attached to them, is entirely confusing. Example. What do you mean by "provide epsilon as the limit"? Classically epsilon \(\varepsilon\) is always used as the name for a fixed but arbitrary positive real number. Not as it is done here, as if it is the name for something that is given a priori and assumed already familiar to the reader.

The author is still being very careless. [1] is taken to mean the version of the weak Goldbach problem in which every odd number \(\geq 7\) is a sum of three primes. This holds for the number 7, by writing 7 as the sum \(2+2+3.\) Sometimes you have to use the even prime 2. Now in equation [5] some of the primes may be even. Therefore it is wrong to say that \(p_6+p_7+1\) is an odd number. And it is also wrong to say that it is at least as large as 7. To get to [7] from [6] is possible if and only if \(p_8=1.\) This is false however. Similarly it is possible to deduce [11] from [10] if and only if \(p_7+p_8=2.\) This is absurd. The logic is very flawed. It is very easy to find primes \(p_1,\ldots, p_6\) for which [9] is true. So it is meaningless to try to deduce a contradiction from [9] alone.

To a child, this might seem like magic. Does it appear to you that this phenomenon instead bears some sort of scientific explanation?

Thank you: I see now which part of the explanation you refer to as "paragraph 1". The confusion comes from the fact that it does not appear to explain the things that come later. My question remains the same. In [8] you say \(p_1+p_2+p_3+p_4 = 2N. \) And (skipping the intermediate [9]) in [10] you say in effect \( N-1 = p_2+p_4.\) Are the primes not supposed to be odd? What if \(N\) is an even number?

Again, what is your paragraph number 1, please explain what it says. If \(2N = 12,\) and \(p_1=p_2=p_3=p_4=3,\) then if \(2N_1=10\) is the previous even number 10 before 12, then \(p_2+p_4 = 3 + 3 = 6 \neq 2N_1 = 10.\) So why do you say that \(p_2+p_4=2N_1?\)

Very well then. But your paragraphs are not numbered. Which one of your paragraphs do you number by 1, what does it say? In [8] what is N? You did not specify what N means. If \(2N_1\) is the "previous even number" (what does that mean?) and [8] says that \(p_1+p_2+p_3+p_4 = 2N\) holds, then why is \(p_2+p_4=2N_1\) true?

How can you take \(p_2\) and \(p_4\) from [8] to use in [9]?

Thanks for the useful link! Regarding an "original theory", such a one would have a theorem like \( dy = \frac{dy}{dx}dx,\) which means that you can do arithmetic with an "infinitesimal" \(dx\), assumed nonzero. In contemporary mathematics the same expression is still a theorem, but it stands for something entirely different; both \(x\) and \(y\) are functions that have differentials \(dx\) and \(dy\) respectively, with \(\frac{dy}{dx}\) being their derivative. None of the latter functions represent anything "infinitely small", indeed the range of either differential can easily be unbounded. In that sense the "theories" somehow should not be considered comparable, because they speak of completely different things. On the other hand, they might be, at least partially, "isomorphic", by being able to show theorems that have an identical outward appearance. I am definitely interested in any small nuances which would make a proposed "original theory" make a different prediction than what we would expect today.

I have read somewhere that Cauchy tried to use the "original theory" in an argument, but ended up with a wrong result. I will look for the reference, though maybe someone knows already?!

Absolutely not. Clearly 150kg is the combined mass of the two objects .

Approaching the question from the mathematical side (and being somewhat naive about the physics), the question isn't necessarily so much about the finiteness of time, but rather whether time has an actual "beginning". Assuming that time will proceed forever after (I have not seen anybody opposed to that assumption), there could be three distinct possibilities: (1) time progresses as \(t \in (-\infty,\infty)\), or (2) as \( t \in (0,\infty)\), or (3) as \( t \in [0,\infty],\) where \(0\) is a 'starting point' for the existence of time. Possibilities (1) and (2) are homeomorphic, in particular, for every time, there was a time prior to it. The only difference being that the progression of time would appear faster in (2) as in (1). Possibility (3) would be similar to the familiar picture of BB in a spacetime with only one space dimension, where BB represents the North Pole of the globe and the time represents moving south along laterals such that space is represented by ever increasing circles. It also means that there were times for which there was no time that was 1 second previously. Does this seem the most reasonable idea of the 'beginning' of time? Oh, and if my intuition is worth much, then I do not see much chance of making any determination of either possibility by physical measurement. But feel free to disagree.

I am not sure what your "it" stands for. In the turtle picture, a turtle is supposed to stand on a lower turtle. I am trying to suggest that a segment of time does not have to stand on any previous segment of time necessarily. Viewed in isolation it could conceivably be the initial segment of time that exists, or it might have any amount of time preceding it. Which is why I would hesitate to entertain the turtle analogy when it comes to time progression. If you mean to point to different possible choices of inertial frame of reference, then the question of finiteness of time appears to be frame independent. I.e. if an amount of time is finite in any frame then it would also measure finite as seen from any other frame, even when the precise length of time measures different.

For the turtles to do proper support, there cannot be a bottom turtle, unless it gets provided with something else to stand on. It doesn't seem that a time measuring device must have a similar property. Either there is a first second (say, after BB), or there isn't one, it doesn't appear to make a big difference to what comes after.